Keywords: Recent Advances in Synchronization (and Observer-Design) for Chaotic or Complex Systems
Abstract: A new finite-dimensional adaptive observer is proposed for some linear parabolic systems. The observer is based on the modal decomposition approach and uses a classical persistent excitation condition to ensure practical exponential convergence of both states and parameters estimation.

Keywords: Recent Advances in Synchronization (and Observer-Design) for Chaotic or Complex Systems
Abstract: The aim of this paper is to provide a novel systematic methodology for the design of sampled-data observers for Linear Kuramoto-Sivashinsky systems (LKS) with non-local outputs. More precisely, we extend the systematic sampled-data observer design approach which is based on the use of an Inter-Sample output predictor to the class of LKS systems. By using a small- gain methodology we provide sufficient conditions ensuring the Input-to-Output Stability property of the estimation errors in the presence of measurement noise. Our Inter-Sample output predictor contains a tuning term which can enlarge significantly the Maximum Allowable Sampling Period.

Keywords: Recent Advances in Synchronization (and Observer-Design) for Chaotic or Complex Systems, New Applications in Chaos Control, Chaos-Encrypted Signals, Optical Systems, Biological Systems, Power Converters, Economic Systems
Abstract: This paper focuses on the interest of using chaotic hyperbolic Partial Differential Equations (PDE) systems synchronization for secure image communication. Such systems offer a promising alternative to finite-dimensional chaotic systems since they enlarge the key space, efficient computation, and more complex dynamics. Here, we modify the classical chaotic hyperbolic PDE system derived from a wave equation with Van der Pol nonlinear boundary condition. The proposed system contains additional integral boundary couplings and potential terms. The chaotic behavior of the resulting system is observed in numerical simulations. Adding such couplings allows for the enhancement of the security of the cryptosystem since it enlarges considerably the key space. The synchronization of receiver and transmitter chaotic systems is achieved by the use of a chaotic observer designed using the backstepping methodology. The initial image can then be encrypted into a chaotic signal on the transmitter side before being sent to the receiver side. Using observational data, the receiver can then reconstruct the initial image. Numerical illustrations complete the paper.

Keywords: Recent Advances in Control and Anti-Control of Chaotic or Complex Systems, Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems, Theory and Applications of Complex Dynamical Networks
Abstract: In a MIMO a nonlinear system having the same number of inputs and outputs, if certain coefficients that characterize its normal form have a special structure, a number of relevant design problems can been successfully addressed. In this paper we consider the more general case of a system having more inputs than outputs, we show that such special structure implies the regularity of the algorithm for construction of a right-inverse, and we show how a recently developed design method for robust control can be implemented on this class of MIMO systems.

Keywords: Recent Advances in Control and Anti-Control of Chaotic or Complex Systems, Bifurcations in Chaotic or Complex Systems, Analysis of Stability, Controllability and Observability of Chaotic or Complex Systems
Abstract: Conditions for designing a feedback control law to remove the dense set of Unstable Periodic Orbits and chaotic attractors while preserving equilibria are formulated in terms of 2-contractivity of the closed loop system, by means of the second additive compound of the system's Jacobian. Matrix inequalities to allow computing the control gain matrix are discussed. An example of application of the method to the Thomas' system is presented.

Keywords: Recent Advances in Control and Anti-Control of Chaotic or Complex Systems, Providing A Discussion Forum for Physics, Chaos and Control System Communities, Bridging the Gap Between Ergodic and Deterministic Approaches
Abstract: Chaos is the most complex dynamic behavior a system can show. Yet, it is produced by two simple, though nonlinear, deterministic mechanisms, hence the more proper name Deterministic Chaos. The first mechanism, stretching, acts locally and linearly, expanding small sets in the system's state space in (at least) one direction and contracting in the others. The second mechanism, folding, acts globally and is the nonlinear operation of bending the elongated sets to form the forward image of the original sets. These two simple operations are invertible and form a chaotic set for the system's forward and backward map, containing an uncountable infinity of astonishingly complex system's trajectories. The simplest explanation of chaos I ever heard is James A. Yorke "pizza machine" (plenary lecture at the SIAM DS, Snowbird, 2003): kneading the pizza dough is nothing but stretching and folding. The mathematical formalization is Stephen Smale's horseshoe map. Using this nice tool, I revisit the hallmarks of chaos, one of which (in the title) still misses a constructive proof.

Keywords: Recent Advances in Control and Anti-Control of Chaotic or Complex Systems, Synchronization of Delay Systems, Chaotic Delay Systems
Abstract: The role of time delay in the control processes aimed at mitigating stick-slip vibration in drill-strings, is of paramount importance. Its influence on the controlled system's settling time cannot be understated; indeed, it can lead to significant consequences such as controller instability. In this study, a numerical analysis was conducted, initially a sliding mode control was employed to suppress stick-slip vibration in drill-string. Then we explore a spectrum of time delay values alongside varying reference angular velocities. The results of this investigation suggest a critical finding: the controller's susceptibility to limit cycles is notably heightened in the presence of time delay, and it experiences a decline when faced with fluctuating reference angular velocities. Despite this degradation, it is noteworthy that the system does not always reach a state of complete instability. These insights are invaluable, offering essential considerations for the development of robust control strategies. Addressing the challenges posed by time delay-induced issues in suppressing stick-slip vibration during drilling operations requires a nuanced approach, taking into account the intricacies highlighted in this study.